Three quadratic equations
Source: INMO 2005 Problem 3
August 23, 2005
quadraticsVietaalgebraalgebra unsolved
Problem Statement
Let be positive real numbers, not all equal, such that some two of the equations \begin{eqnarray*} px^2 + 2qx + r &=& 0 \\ qx^2 + 2rx + p &=& 0 \\ rx^2 + 2px + q &=& 0 . \\ \end{eqnarray*} have a common root, say . Prove that is real and negative; the remaining third quadratic equation has non-real roots.